Such a cutting unit is known, for example, from U.S. Pat. No. 3,890,705 . The dams are obtained, for example, in that the slot-shaped hair entrance apertures are sawn in a basic form of sheet material of the cutting head. During shaving, a hair which is in a hair entrance aperture is cut by the cooperation of a dam and the cutting member. The thinner the dam is constructed, the closer to the skin will a hair be cut and the better hence is the shaving result. The dimensions of a dam, in particular the thickness, however, cannot be chosen to be arbitrarily small because the dam must have sufficient rigidity, for example, for providing the reaction force necessary during cutting and to prevent the dam from bending too far inwardly as a result of the pressure upon contacting the skin to be shaved.